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Finding the minimizer

  1. Jul 7, 2011 #1
    I am trying to find the minimizer of the function

    [itex]\left\| \nabla f(x) + \lambda ^T \nabla h(x) + \mu ^T \right\|^2[/itex]
    s.t. [itex]\mu _i \geq 0[/itex] , [itex]\mu _i = 0[/itex] if [itex]x_i > 0[/itex]

    We use the function [itex]\phi _i (\mu ) = min \left\{ \mu _i , x_i \right\}[/itex]
    we have that [itex]\mu ^T x = 0 \Leftrightarrow \phi ( \mu ) =0[/itex]

    So we can actually solve the problem

    Minimize [itex]\left\| \nabla f(x) + \lambda ^T \nabla h(x) + \mu ^T \right\|^2 + \left\| \phi (\mu ) \right\|^2 [/itex]
    s.t. [itex]\lambda , \mu \geq 0[/itex]

    Now my reasoning is, by letting [itex]g= \nabla f(x) + \lambda ^T \nabla h(x) [/itex] , the problem becomes:

    Minimize [itex] (g+ \mu ) ^2 + \phi (\mu ) ^2 [/itex] i.e. Minimize [itex] (g+ \mu ) ^2 + min \left\{ \mu , x \right\}^2 [/itex]
    or Minimize [itex] (g^2+ 2g \mu +\mu ^2 ) + \left\{ \mu ^2 or \quad x^2 \right\} [/itex]

    Now, I think I should first find the critical points for the function [itex] (g^2+ 2g \mu +\mu ^2 ) + \left\{ \mu ^2 or \quad x^2 \right\} [/itex] . But, should I consider this function as a function of [itex] \mu [/itex] and [itex]g[/itex], or as a function of [itex]\mu [/itex]and [itex]x [/itex]?

    Another way of thinking about this problem. If I use the 'Fischer-Burmeister' function which is:

    I am trying to find the minimizer of the function

    [itex]\left\| \nabla f(x) + \lambda ^T \nabla h(x) + \mu ^T \right\|^2[/itex]
    s.t. [itex]\mu _i \geq 0[/itex] , [itex]\mu _i = 0[/itex] if [itex]x_i > 0[/itex]

    We use the function [itex]\phi _i (\mu ) = min \left\{ \mu _i , x_i \right\}[/itex]
    we have that [itex]\mu ^T x = 0 \Leftrightarrow \phi ( \mu ) =0[/itex]

    So we can actually solve the problem

    Minimize [itex]\left\| \nabla f(x) + \lambda ^T \nabla h(x) + \mu ^T \right\|^2 + [itex]\left\| \phi (\mu ) \right\|^2 [/itex]
    s.t. [itex]\lambda , \mu \geq 0[/itex]

    Now my reasoning is, by letting [itex]g= \nabla f(x) + \lambda ^T \nabla h(x) [/itex] , the problem becomes:

    Minimize [itex] (g+ \mu ) ^2 + \phi (\mu ) ^2 [/itex] i.e. Minimize [itex] (g+ \mu ) ^2 + min \left\{ \mu , x \right\}^2 [/itex]
    or Minimize [itex] (g^2+ 2g \mu +\mu ^2 ) + \left\{ \mu ^2 , x^2 \right\} [/itex]

    Now, I think I should first find the critical points for the function [itex] (g^2+ 2g \mu +\mu ^2 ) + \left\{ \mu ^2 , x^2 \right\} [/itex] . But, should I consider this function as a function of [itex] \mu [/itex] and [itex]g[/itex], or as a function of [itex]\mu [/itex]and [itex]x [/itex]?

    Another way of thinking about this problem. We can use the 'Fischer-Burmeister' function which is:

    [itex]\Phi (\mu , x ) = \mu + x - \sqrt{\mu ^2 + x^2} [/itex] instead of the function [itex]\phi _i (\mu ) = min \left\{ \mu _i , x_i \right\}[/itex] because for the 'Fischer-Burmeister' function,
    [itex]\Phi (\mu , x ) = 0 \Leftrightarrow \mu x =0 [/itex] just like for the previous function [itex]\phi _i (\mu )[/itex] .

    Now, the problem would be to

    Minimize [itex] (g+ \mu ) ^2 + \Phi (\mu ) ^2 [/itex] .

    Again, how should I go about finding this minimum?
     
  2. jcsd
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